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  "body": "\n## Summary\n\nThe Bargmann transform is a unitary map from $L^2(\\R)$ onto the Segal–Bargmann space, the Hilbert space of [[entire functions|0G4F0XB9X2]] $F$ on $\\C$ with $\\int_\\C \\abs{F(z)}^2 e^{-\\abs{z}^2}\\, dA(z) \u003c \\infty$.\nIt sends the [[Hermite functions|52VS07H6GK]] to the normalized monomials $z^n / \\sqrt{n!}$.\nIt turns the creation and annihilation operators of the harmonic oscillator into multiplication by $z$ and differentiation in $z$.\nThis makes the oscillator very simple.\nIt is closely related to the short-time Fourier transform with a Gaussian window.\n\n## Explanations\n\n### L1 human\n\nA function on the real line is turned into a holomorphic function on the complex plane.\nThe real part of $z$ plays the role of position and the imaginary part the role of momentum, so the transform shows a quantum state in phase space.\nThe Gaussian $e^{-x^2/2}$, the ground state of the harmonic oscillator, becomes the constant function $1$.\nHigher energy states become the powers $z^n$, up to a factor.\n",
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